#499760

order	512
nilpotency class	4
isoclinism	isoclinic class of #499757
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2929 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1270 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 68 points:

(5,6)(7,20)(8,25)(9,13)(10,18)(11,17)(12,19)(14,23)(15,26)(16,30)(21,50)(22,45)(24,47)(27,46)(28,58)(29,59)(31,40)(32,42)(33,37)(34,38)(35,41)(36,63)(39,64)(43,61)(44,62)(48,51)(49,54)(52,57)(53,68)(55,67)(56,65)(60,66);(1,2)(3,4)(6,13)(9,30)(10,11)(14,36)(15,25)(16,18)(17,37)(19,38)(20,45)(22,54)(23,35)(26,32)(27,28)(31,63)(33,34)(39,57)(40,47)(41,49)(42,46)(43,58)(44,59)(48,50)(51,55)(52,56)(60,64)(61,62)(65,67)(66,68);(2,4)(5,7)(6,14)(8,21)(9,31)(10,35)(11,23)(12,24)(13,36)(15,39)(16,45)(17,47)(18,20)(19,41)(22,34)(25,57)(26,60)(27,56)(28,52)(29,53)(30,63)(32,64)(33,54)(37,40)(38,49)(42,48)(43,66)(44,67)(46,50)(51,62)(55,61)(58,68)(59,65);(5,8,12,29)(6,15,19,44)(7,21,24,53)(9,32,34,61)(10,28,11,27)(13,25,38,59)(14,39,41,67)(16,46,37,58)(17,43,18,42)(20,48,47,66)(22,55,31,64)(23,56,35,52)(26,33,62,30)(36,57,49,65)(40,68,45,50)(51,63,60,54)

Polycyclic presentation:

2^9|d^2=i,e^2=hi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,h^a=hi,c^b=cg,e^b=eh,f^b=fi,h^b=hi,e^c=ei,f^c=fi,e^d=eh,f^d=fi,h^d=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406956);